Why are state-values alone not sufficient in determining a policy (without a model)?
why is it not possible to suggest a policy solely on the basis of state-values; why do we need state-action values?

A policy function takes state as an argument and returns an action $a = \pi(s)$, or it may return a probability distribution over actions $\mathbf{Pr}\{A_t=a|S_t=s \} =\pi(a|s)$.
In order to do this rationally, an agent needs to use the knowledge it has gained to select the best action. In value-based methods, the agent needs to identify the action that has the highest expected return. As an aside, whilst learning it may not take that action because it has decided to explore, but if it is not capable of even identifying a best action then there is no hope of it ever finding an optimal policy, and it cannot even perform $\epsilon$-greedy action selection, which is a very basic exploration approach.
If you use an action value estimate, then the agent can select the greedy action simply:
$$\pi(s) = \text{argmax}_a Q(s,a)$$
If you have state values, then the agent can select the greedy action directly only if it knows the model distribution $p(r,s'|s,a)$:
$$\pi(s) = \text{argmax}_a \sum_{r,s'}p(r,s'|s,a)(r + \gamma V(s'))$$
In other words, to find the best action to take the agent needs to look ahead a time step to find out what the distribution of next states would be following that action. If the only values the agent knows are state values, this is the only way the agent can determine the impact of any specific action.
Although there are alternatives to this specific equation, there is no alternative that does not use a model in some form. For instance, if you can simulate the environment, you could simulate taking each action in turn, and look over multiple simulation runs to see which choice ends up with the best $(r + \gamma V(s'))$ on average. That would be a type of planning, and perhaps the start of a more sophisticated approach such as MCTS. However, that simulation is a model - it needs access to the transition probabilities in some form in order to correctly run.
It is possible to have an entirely separate policy function that you train alongside a state value function. This is the basis of Actor-Critic methods, which make use of policy gradients to adjust the policy function, and one of the value-based methods, such as TD learning, to learn a value function that assists with calculating the updates to the policy function. In that case you would not be using a value-based method on its own, so the quote from that part of Sutton &amp; Barto does not apply.